docs(knowledge): §7 + E-WHP-BIPOLAR-1 — bipolar phase = Walsh-Hadamard on VSA (policy mirror)

Policy-side mirror of the operator pin: signed phase makes the cascade
a Walsh-Hadamard transform on the existing VSA-bipolar algebra. Signs
compose by XOR (= vsa_bind); magnitudes compose by vsa_bundle
(Markov-respecting per I-SUBSTRATE-MARKOV). Each cell = Walsh-resonance
superposition recoverable by role-key unbind; I-VSA-IDENTITIES Test 1
(N <= sqrt(d)/4 ~= 32) IS the substrate uncertainty principle made
explicit. Roundtrip bit-exact (phase generated, Walsh self-inverse).

TWO-ALGEBRA RULE (load-bearing): sign = XOR; magnitude = bundle, NEVER
MergeMode::Xor (breaks Markov; PP-13 P1-1 is the named anti-pattern
that conflates them). Receipts already exist: Vsa16kF32 (bipolar
carrier, crystal/fingerprint.rs), vsa_bind/vsa_bundle (iron-rule
algebra), helix CurveRuler (bit-exact integer phase walk).

Fences: 'quantum-like' is the bundling algebra NOT measurement
randomness (no headline drift); bipolar = 1-bit phase (multi-bit stacks
above when needed); Parseval requires the bundle (sum), not just XOR.

Probes shared with ndarray §4b: WHP-1 (Parseval rho on random
sign-fields, Jirak floor) / WHP-2 (cross-backend roundtrip
byte-identical) / WHP-3 (bind+unbind margin holds for N <= sqrt(d)/4,
fails cleanly past it) / WHP-4 (two-algebra guard: raw-XOR on
magnitudes fails Chapman-Kolmogorov consistency).

EPIPHANIES E-WHP-BIPOLAR-1 prepended (board hygiene).

https://claude.ai/code/session_01PBTGaPCSnnt6u3pjXpbLwY

Author: claude

.claude/board/EPIPHANIES.md

@@ -49,6 +49,37 @@ They are the SAME relation: the north-star template hierarchy IS the routing/sup
 **Phase B becomes:** stand up OGAR as the OGIT mirror + north-star template registry; seed entity_type↔NiblePath from it; the build-time round-trip proves the bijection. The surrealdb-coords blocker (N8 / Phase H) is unrelated and remains.
 
 **Cross-ref:** identity-architecture plan DECISION-2 + the north-star guard; E-IDENTITY-WHITEBOX-1 (NodeGuid composition); I-VSA-IDENTITIES (closed template vocabulary interns; Wikidata's open instance mass stays content, never a ClassId).
+## 2026-06-10 — E-WHP-BIPOLAR-1 — bipolar phase makes the perturbation pyramid a Walsh-Hadamard transform on VSA (deterministic, quantum-shaped, classical)
+
+**Status:** FINDING (operator-pinned in `OGAR/CLAUDE.md`; crystallized both sides).
+**Confidence:** High that receipts already exist (VSA bind/bundle is the iron-rule
+algebra; `Vsa16kF32` is bipolar; helix `CurveRuler` is the bit-exact integer phase
+generator); CONJECTURE on synthesis-as-Walsh-pyramid until WHP-1..4 land.
+
+When the deterministic phase from the §6 perturbation pin is made **signed (±1)** —
+one bit per (addr, level) — the cascade IS the Walsh-Hadamard transform of the
+address tree, carried on the workspace's existing VSA-bipolar algebra: signs compose
+by XOR (= `vsa_bind`), magnitudes compose by `vsa_bundle` (Chapman-Kolmogorov-
+respecting per `I-SUBSTRATE-MARKOV`). Each cell is a Walsh-resonance superposition
+recoverable by role-key unbind; `I-VSA-IDENTITIES` Test 1 (N ≤ √d/4 ≈ 32) IS the
+substrate's uncertainty principle. **Roundtrip bit-exact** because phase is
+generated from the address, not stored — Walsh-Hadamard is self-inverse up to
+scale. "Schrödinger's cat in a glass box": superposition is real, identity
+recoverable by key, no measurement randomness.
+
+**TWO-ALGEBRA RULE (load-bearing):** sign = XOR; magnitude = bundle, NEVER
+`MergeMode::Xor` (breaks Markov; the named anti-pattern is PP-13 P1-1 "raw-XOR
+ordering as 'nearest'"). Sign side preserves the write-back data-flow rule
+(single-target gated XOR is allowed); magnitude side preserves Parseval
+(L2 conservation → "top gaussian preserved", not Kombinatorik-style selection).
+
+Honest fences: "quantum-like" is the bundling algebra, not measurement randomness
+(no headline drift); bipolar = 1-bit phase (multi-bit stacks above when measured to
+be needed); Parseval requires the bundle, not just XOR. Probes WHP-1..4 land before
+any L2-conservation claim ships. Full treatment: ndarray
+`guid-prefix-shape-routing.md` §4b; policy mirror:
+`guid-canon-and-prefix-routing.md` §7; canon: `OGAR/CLAUDE.md`.
+
 ## 2026-06-10 — E-CANON-GUID-1 — the canonical GUID's dash-groups are the cascade; routing/quorum crystallized before dilution
 
 **Status:** FINDING (canon operator-pinned in `OGAR/CLAUDE.md`; crystallization docs landed both sides)

.claude/knowledge/guid-canon-and-prefix-routing.md

@@ -87,5 +87,36 @@ phase is convention, not data; the analysis remainder escalates to
 full-residual per the quorum certificate, never silently. Added
 probes: **PROBE-PHASE-1** (bit-exact cross-backend regeneration;
 integer walk only per D-QUANTGATE) · **PROBE-PERT-RHO** (magnitude-only
-meets ρ anchors; escalation rate reported). Full treatment: ndarray
+meets ρ anchors; escalation rate reported). Bipolar sharpening (§7
+below). Full treatment: ndarray
 `guid-prefix-shape-routing.md` §4; canon: `OGAR/CLAUDE.md`.
+
+## 7. Bipolar-phase pyramid = Walsh-Hadamard on VSA (operator, 2026-06-10)
+
+Sharper form of §6: when the deterministic phase is **signed (±1)**,
+the cascade IS the Walsh-Hadamard transform of the address tree,
+carried on the workspace's existing `Vsa16kF32` bipolar algebra.
+**Signs compose by XOR (= `vsa_bind`)**; **magnitudes compose by
+`vsa_bundle`** (Markov-respecting per `I-SUBSTRATE-MARKOV`). Each cell
+is a Walsh-resonance superposition recoverable by role-key unbind;
+the Heisenberg-shaped capacity bound is `I-VSA-IDENTITIES` Test 1
+(N ≤ √d/4 ≈ 32) — the classical uncertainty principle of the substrate
+made explicit. Roundtrip bit-exact because phase is generated, not
+stored.
+
+**Two-algebra rule (load-bearing):** sign = XOR; magnitude = bundle,
+NEVER `MergeMode::Xor` (breaks Markov; the named anti-pattern is PP-13
+P1-1, "raw-XOR ordering as 'nearest'"). Receipts already exist:
+`vsa_bind` / `vsa_bundle` (iron-rule algebra), `Vsa16kF32` (bipolar
+carrier in `crystal/fingerprint.rs`), helix `CurveRuler` (bit-exact
+integer phase walk, D-QUANTGATE-compliant). Honest fences:
+"quantum-like" is the bundling algebra, NOT measurement randomness;
+bipolar = 1-bit phase; Parseval preservation requires the bundle (sum),
+not just XOR.
+
+**Probes added:** **WHP-1** (Parseval: `Σ|cell|² = Σ|M_L|²` within
+Jirak noise floor) · **WHP-2** (encode→decode→encode byte-identical
+cross-backend) · **WHP-3** (bind/unbind margin holds for N ≤ √d/4;
+fails cleanly past it) · **WHP-4** (explicit two-algebra guard:
+raw-XOR on magnitudes fails Chapman-Kolmogorov consistency). Full
+treatment: ndarray `guid-prefix-shape-routing.md` §4b.